Final answer:
To evaluate the integral in Gauss's law, identify the symmetry of the charge distribution, choose a Gaussian surface with the same symmetry, calculate the electric flux through the surface by evaluating the integral of the electric field, and determine the total enclosed charge divided by the permittivity of free space.
Step-by-step explanation:
The integral in Gauss's law should be evaluated by following a specific strategy. First, identify the spatial symmetry of the charge distribution to choose an appropriate Gaussian surface that has the same symmetry. Then, calculate the electric flux through the chosen Gaussian surface by evaluating the integral of the electric field over the surface. This integral can be computed by taking the antiderivatives of the dimensions defining the surface area. Finally, determine the total charge enclosed within the Gaussian surface and divide it by the permittivity of free space to obtain the flux through the closed surface.